
// Make newform 6171.2.a.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6171_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6171_2_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_6171_2_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_6171_a();"
function MakeCharacter_6171_a()
    N := 6171;
    order := 1;
    char_gens := [4115, 970, 2179];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_6171_a_Hecke(Kf)
    return MakeCharacter_6171_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0], [1], [3], [4], [0], [1], [1], [1], [9], [-6], [2], [-4], [3], [7], [-6], [-6], [6], [-8], [-4], [12], [-2], [10], [6], [0], [-16], [0], [5], [-9], [-20], [-9], [13], [-3], [-6], [-2], [18], [-8], [11], [2], [-21], [-15], [-6], [14], [18], [22], [-3], [-16], [-2], [-1], [-3], [14], [-21], [12], [-8], [24], [12], [-12], [-15], [-11], [-2], [-12], [10], [24], [-20], [-24], [-16], [-6], [-13], [-14], [12], [19], [6], [0], [8], [22], [32], [-12], [36], [20], [-15], [19], [12], [-25], [24], [-1], [28], [6], [-6], [19], [-30], [8], [42], [27], [-22], [6], [-22], [-15], [36], [21], [-20], [16], [-8], [30], [30], [24], [-8], [-7], [-36], [18], [-6], [-38], [-38], [-11], [-6], [-10], [-37], [-33], [32], [18], [27], [-6], [-31], [22], [27], [3], [20], [-24], [14], [3], [8], [-2], [1], [-24], [-46], [-43], [-30], [-41], [24], [40], [-48], [51], [10], [21], [-28], [21], [14], [-57], [-26], [18], [20], [24], [-14], [18], [11], [-39], [2], [-27], [-11], [15], [22], [-18], [36], [-36], [-41], [6], [-30], [-9], [-52], [-62], [-20], [6], [-42], [5], [-9], [-59], [-61], [51], [-20], [-15], [43], [-47], [-19], [60], [-22], [-36], [-18], [54], [-38], [-19], [-62], [-36], [-10], [12], [-1], [48], [9], [6], [34], [44], [-42], [54], [15], [28], [-7], [-17], [-60], [54], [-40], [48], [27], [20], [-62], [54], [26], [-12], [-39], [-28], [43], [36], [-45], [-48], [7], [-68], [48], [14], [18], [34], [-69], [57], [-41], [-48], [-1], [70], [-20], [66], [56], [42], [-40], [-33], [-15], [-12], [21], [-20], [-49], [11], [-51], [-9], [-10], [-21], [19], [36], [-41], [-18], [48], [-64], [-42], [-60], [-16], [52], [51], [58], [10], [-48], [-68], [40], [9], [-37], [-66], [0], [64], [18], [32], [-55], [-49], [-32], [-14], [-25], [36], [43], [19], [0], [-60], [50], [72], [10], [22], [48], [74], [-42], [35], [0], [27], [-3], [66], [-3], [64], [-63], [80], [-48], [-33], [-68], [31], [24], [4], [9], [-28], [2], [42], [-4], [-60], [22], [12], [-18], [-3], [59], [-48], [-23], [-30], [-3], [68], [-36], [46], [-22], [-66], [-13], [-54], [-61], [2], [-10], [-24], [-45], [28], [81], [-62], [42], [16], [-75], [65], [81], [38], [-29], [-19], [-12], [-51], [71], [24], [51], [2], [47], [-57], [-42], [-5], [-4], [-36], [10], [49], [0], [-36], [84], [-3], [-30], [-44], [-81], [-24], [-36], [-43], [17], [75], [52], [-8], [12], [-76], [57], [0], [40], [8], [-45], [-36], [49], [90], [-17], [15], [96], [-44], [-15], [70], [93], [17], [38], [-56], [96], [-22], [-87], [-12], [-1], [-6], [4], [-104], [6], [20], [18], [64], [-48], [31], [30], [6], [-62], [14], [-60], [77], [105], [-44], [78], [0], [28], [-65], [-30], [15], [14], [63], [30], [6], [-98], [3], [78], [-64], [24], [48], [27], [65], [9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6171_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6171_2_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_6171_2_a_e(:prec:=1)
    chi := MakeCharacter_6171_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_6171_2_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_6171_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6171_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<5,R![-3, 1]>,<7,R![-4, 1]>],Snew);
    return Vf;
end function;